584 research outputs found
Chaotic transport in the homoclinic and heteroclinic tangle regions of quasiperiodically forced two-dimensional dynamical systems
The authors generalize notions of transport in phase space associated with the classical Poincare map reduction of a periodically forced two-dimensional system to apply to a sequence of nonautonomous maps derived from a quasiperiodically forced two-dimensional system. They obtain a global picture of the dynamics in homoclinic and heteroclinic tangles using a sequence of time-dependent two-dimensional lobe structures derived from the invariant global stable and unstable manifolds of one or more normally hyperbolic invariant sets in a Poincare section of an associated autonomous system phase space. The invariant manifold geometry is studied via a generalized Melnikov function. Transport in phase space is specified in terms of two-dimensional lobes mapping from one to another within the sequence of lobe structures, which provides the framework for studying several features of the dynamics associated with chaotic tangles
Persistence of instanton connections in chemical reactions with time dependent rates
The evolution of a system of chemical reactions can be studied, in the
eikonal approximation, by means of a Hamiltonian dynamical system. The fixed
points of this dynamical system represent the different states in which the
chemical system can be found, and the connections among them represent
instantons or optimal paths linking these states. We study the relation between
the phase portrait of the Hamiltonian system representing a set of chemical
reactions with constant rates and the corresponding system when these rates
vary in time. We show that the topology of the phase space is robust for small
time-dependent perturbations in concrete examples and state general results
when possible. This robustness allows us to apply some of the conclusions on
the qualitative behavior of the autonomous system to the time-dependent
situation
Splitting of separatrices, scattering maps, and energy growth for a billiard inside a time-dependent symmetric domain close to an ellipse
We study billiard dynamics inside an ellipse for which the axes lengths are
changed periodically in time and an -small quartic polynomial
deformation is added to the boundary. In this situation the energy of the
particle in the billiard is no longer conserved. We show a Fermi acceleration
in such system: there exists a billiard trajectory on which the energy tends to
infinity. The construction is based on the analysis of dynamics in the phase
space near a homoclinic intersection of the stable and unstable manifolds of
the normally hyperbolic invariant cylinder , parameterised by the
energy and time, that corresponds to the motion along the major axis of the
ellipse. The proof depends on the reduction of the billiard map near the
homoclinic channel to an iterated function system comprised by the shifts along
two Hamiltonian flows defined on . The two flows approximate the
so-called inner and scattering maps, which are basic tools that arise in the
studies of the Arnold diffusion; the scattering maps defined by the projection
along the strong stable and strong unstable foliations of the
stable and unstable invariant manifolds at the homoclinic
points. Melnikov type calculations imply that the behaviour of the scattering
map in this problem is quite unusual: it is only defined on a small subset of
that shrinks, in the large energy limit, to a set of parallel lines
as .Comment: 25 page
Homoclinic Orbits In Slowly Varying Oscillators
We obtain existence and bifurcation theorems for homoclinic orbits in three-dimensional flows that are perturbations of families of planar Hamiltonian systems. The perturbations may or may not depend explicitly on time. We show how the results on periodic orbits of the preceding paper are related to the present homoclinic results, and apply them to a periodically forced Duffing equation with weak
feedback
Horseshoes and Arnold Diffusion for Hamiltonian Systems on Lie Groups
Abstract not availabl
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